Atlas › Modular forms › Weight 2 › Level 1000 › 1000.2.a › 1000.2.a.f

Newform orbit 1000.2.a.f

Trace form

The 4 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$4q - q^{3} - 9q^{7} + 11q^{9} + 5q^{11} + 4q^{13} + 4q^{17} + 3q^{21} - 11q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 4. It is generated by the eigenvalue $\alpha$ of $T = T_{3} + 3T_{7}$ , whose minimal polynomial is

$x^{4} + 28x^{3} + 169x^{2} - 628x - 4999$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 1000$ )
5	— ( $5^2 \mid 1000$ )

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	−1	0	−9	5	4	4	0	−11	10	9	15	17	12	−14	−6	18	11	−1	26	9	−8	12	5	29

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.498 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound. The label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000]: [0, 1, 2, 1, 3, 3, 6, 3, 10, 9, 18, 13, 34, 39, 78, 154] (certified at ell = 2147482801)
T = [(3, 1), (7, 3)]; h of degree 24 = 1 x 24 (phi(ord eps) x dim S_k^new); 4 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(1000, 2)[5]   # 1000.2.a.f

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 1000:2 1000